We present errors of quadrature rules for the nearly singular integrals in the momentum-space bound-state equations and give the critical value of the nearly singular parameter. We give error estimates for the expansion method, the Nyström method, and the spectral method which arise from the near singularities in the momentum-space bound-state equations. We show the relations amongst the near singularities, the odd phenomena in the eigenfunctions, and the unreliability of the numerical solutions.